Topological Surgery in Lorentzian Geometry

• Topological surgery in Lorentzian geometry is a process that modifies spacetime topology via handle attachment, blending Morse theory with Lorentzian cobordism in 4-manifolds.
• The methodology integrates handlebody theory, explicit metric construction, and strict topological invariants to model processes like wormhole nucleation and cosmic string collapse.
• Applications of this framework include modeling singularity resolution, traversable wormholes, and causal structure in classical and quantum gravitational contexts.

Topological surgery in Lorentzian geometry provides a precise mathematical framework for describing topology change in spacetime, such as the nucleation of wormholes and the evolution of black holes. It synthesizes handlebody theory, Morse theory, and Lorentzian cobordism, facilitating rigorous treatments of smooth changes in spatial topology embedded within globally hyperbolic Lorentzian (or, more generally, $\Spin(1,n)_0$-structured) 4-manifolds. The central questions are: which topology-changing processes are permitted within general relativity or quantum gravity, and how can explicit metrics realizing these transitions be constructed? This entry reviews both the topological foundations and the Lorentzian differential-geometric constructions that underlie this field, with close attention to recent models of singularity resolution, cobordism selection rules, and applications in two and four-dimensional gravity.

1. Topological Surgery: Definitions and Embedding into Lorentzian Geometry

Topological surgery is a process by which a manifold $M^n$ is modified via the removal of an embedded submanifold $S^k\times D^{n-k}$ and the attachment of $D^{k+1}\times S^{n-k-1}$ along their boundaries. For spatial 3-manifolds ($M^3$), the principal cases in physical applications are 0-surgery (wormhole formation) and 1-surgery (cosmic string collapse or knot surgery) (Antoniou et al., 2018, Antoniou et al., 2018). Specifically:

• 0-surgery: Remove two disjoint 3-balls ($S^0\times D^3$), glue in a handle $D^1\times S^2$, forming a wormhole throat connecting the excised regions.
• 1-surgery: Remove a solid torus ($S^1\times D^2$), glue in $D^2\times S^1$ via a meridian-longitude identification, modeling the collapse and transition of knotted cosmic strings.

Embedding this process into Lorentzian geometry, one considers a globally hyperbolic 4-manifold $V^4$ with a Cauchy foliation by 3-manifolds $M^3_t$. Surgery is effected by introducing a 4-dimensional handle, transverse to the foliation: at times $t<t^*$, $M^3_t$ contains the excised region; at $t>t^*$, the new handle is in place; at $t^*$, the surgery occurs at a Morse critical point. The entire process is localized in a compact spacetime domain (Antoniou et al., 2018, Pisana et al., 4 May 2025).

2. Lorentzian Cobordism, Surgery, and Obstructions

A Lorentzian cobordism $(M; N_1, N_2)$ consists of a compact $(n+1)$-manifold with boundary $N_1\sqcup N_2$, a Lorentzian metric $g$ with a globally defined timelike line field $V$, such that $N_1$, $N_2$ are spacelike. For topology change to be realized by surgery in Lorentzian signature, there are strict topological restrictions. Smirnov and Torres (Smirnov et al., 2018) show that the key obstructions (in the $\Spin(1,n)_0$ category) are:

• For even $n$: Euler characteristic equality, $\chi(N_1) = \chi(N_2)$.
• For $n \equiv 1,3,5 \pmod{8}$: Kervaire semi-characteristic equality, $\chi_{1/2}(N_1)=\chi_{1/2}(N_2)$ mod 2.
• For $n \equiv 7 \pmod{8}$: no obstruction. If the obstruction vanishes, any Spin cobordism can be modified by interior connected sums with Spin–null–bordisms (e.g., $S^p \times S^{n+1-p}$) to realize a Lorentzian cobordism (Smirnov et al., 2018).

The "gravitational kink number" counts the (signed) intersections of the global timelike direction $V$ with the boundary's inward normal, imposing a mod 2 selection rule linking Lorentzian cobordisms and topological invariants. This structure ensures that only certain surgery-induced topology changes are compatible with Lorentzian spacetime and prescribed structure groups.

3. Explicit Constructions: Wormhole Nucleation and Singularity Resolution

Recent advances provide concrete Lorentzian metrics realizing surgery-induced topology change. The construction of wormhole nucleation via Lorentzian 0-surgery uses Morse theory on a compact 4-manifold $W$ with Morse function exhibiting an index-1 critical point. Near the critical point $p$, a metric is constructed such that (Pisana et al., 4 May 2025):

$f(x) = -\,(x^0)^2 + (x^1)^2 + (x^2)^2 + (x^3)^2,$

with a degenerate Lorentzian metric singular at $p$. To desingularize, one removes a 4-ball around $p$ and forms a connected sum with $\overline{\mathbb{CP}}^2$, in which the singularity is replaced by a compact region containing closed timelike curves (CTCs), while preserving a smooth Lorentzian metric elsewhere. The resulting spacetime violates all standard energy conditions (NEC/WEC) in the CTC region and ensures the finiteness of curvature invariants (Pisana et al., 4 May 2025).

Surgery type	Initial data	Topological change	Lorentzian outcome / remarks
0-surgery	$M^3$, remove $S^0\times D^3$	Attach $D^1\times S^2$ (handle)	Wormhole nucleation; CTCs, NEC violation
1-surgery	$M^3$, remove $S^1\times D^2$	Attach $D^2\times S^1$	Cosmic-string collapse; black hole

The initial and final spatial slices differ by an $S^1\times S^2$ summand: e.g., $S_i = \mathbb{R}^3$, $S_f = \mathbb{R}^3\#(S^1\times S^2)$.

4. Topology Change in 2D Lorentzian (JT) Gravity and Crotch Singularities

In lower-dimensional settings, specifically Jackiw-Teitelboim (JT) gravity, topological surgery appears as a genus expansion in the Lorentzian path integral (Usatyuk, 2022, Blommaert et al., 2023). The process is implemented via isolated metric degeneracies—co-dimension-2 crotch singularities—localized "branch points" with delta-function curvature. Semiclassical wormhole solutions are constructed by inserting crotches at extremal surfaces in Lorentzian metrics:

$$ds^2 = (x^2 + y^2)(dx^2 + dy^2) - (2\pm i\epsilon)(x\ dx - y\ dy)^2,\quad \epsilon\to 0^+,$$

with curvature concentrated at $(x,y) = 0$: $\sqrt{g}\,R = -4\pi\,\delta^{(2)}(x,y)$. This technique provides a real-time Lorentzian realization of Euclidean wormholes and replica geometries, preserving all known semiclassical phenomena such as the ramp and plateau in the spectral form factor (Blommaert et al., 2023).

These singularities act as "lightcone diagrams" where spatial $S^1$ splits or joins, corresponding to the surgical creation or annihilation of baby universes or higher genus handles.

5. Causal Structure, Energy Conditions, and Global Topology

A central feature of topology change in Lorentzian geometry is the interplay between causal continuity, energy conditions, and the presence of singularities or CTC regions. Surgery-induced topology change, even when explicitly constructed via handle attachment or connected sum, generically results in either:

• The formation of a singular region (Morse critical point).
• The appearance of a compact region with CTCs after desingularization (e.g., via the Misner trick). This is unavoidable by the Geroch theorem: nonsingular topology change in globally hyperbolic Lorentzian manifolds is forbidden unless CTCs or NEC violation are present (Pisana et al., 4 May 2025).

For classical black hole or cosmic string scenarios, the surgical region is typically "hidden" behind an event horizon, with the global hyperbolicity and causality of the exterior maintained (Antoniou et al., 2018). Traversable wormholes and nonsingular topology-changing cobordisms require stress-energy tensors violating the null energy condition, often of Hawking–Ellis type IV near the core (Pisana et al., 4 May 2025).

6. Generalizations, Classification, and Open Problems

Topological surgery applies in arbitrary spacetime dimension, with the existence and classification of Lorentzian cobordisms governed by stable invariants (Euler characteristic, Kervaire semi-characteristic) and structure group requirements ($\Spin(1,n)_0$) (Smirnov et al., 2018). The Lorentzian cobordism groups for low dimensions can be computed explicitly, with isomorphism invariants matching the corresponding obstructions.

In 3+1 dimensions, all closed orientable 3-manifolds arise by sequences of knot (1-) surgeries, with the process classified by the underlying knot and framing (Wallace–Lickorish theorem). The ER=EPR conjecture and the dynamics of entangled black hole pairs can be modeled topologically via 0- and 1-surgeries (Antoniou et al., 2018). In 2D (JT gravity), the genus expansion gives a real-time path integral basis for sum-over-topologies, with surgery points represented by crotch singularities (Blommaert et al., 2023, Usatyuk, 2022).

Open problems include the extension of Lorentzian surgery constructions with explicit metrics, junction conditions, and horizon existence theorems in higher dimensions; the physical interpretation and quantum stability of CTC-containing regions; and the development of analogs of "lightcone diagrams" in four-dimensional quantum gravity (Usatyuk, 2022).

7. Summary Table: Key Topological and Geometric Ingredients

Concept	Mathematical Structure	Physical Role	References
$n$-Surgery	Remove $S^k \times D^{n-k}$, glue $D^{k+1} \times S^{n-k-1}$	Topology change (wormholes, black holes)	(Antoniou et al., 2018, Antoniou et al., 2018)
Lorentzian cobordism	$(M; N_1, N_2)$ with global timelike line field	Interpolates spatial topologies over time	(Smirnov et al., 2018)
Morse function/handlebody	Singular point ($\nabla f = 0$), $k$-handle attachment	Local model for surgery in spacetime	(Pisana et al., 4 May 2025)
Connected sum ($\#$)	Remove $D^{n}$, glue along $S^{n-1}$	Desingularization, CTC region introduction	(Pisana et al., 4 May 2025)
Crotch singularity (2D/AdS)	Co-dimension-2 delta curvature, branch point	Real-time wormhole creation, genus expansion	(Blommaert et al., 2023, Usatyuk, 2022)
Cobordism invariants	Euler characteristic, Kervaire semi-characteristic	Selection rules for topology change	(Smirnov et al., 2018)

Topological surgery in Lorentzian geometry thus provides a unified vocabulary and framework for explicit models of topology change, buttressed by deep results from handlebody theory, cobordism classification, and differential topology. Lorentzian constructions require careful attention to causal structure, energy conditions, and singularity resolution mechanisms, and recent developments continue to refine both mathematical understanding and physical interpretation in contexts ranging from classic general relativity to holographic and two-dimensional gravity.